Question

Use the image to answer the question.

A graphed function starting on the x axis has an arrow at the other end passes through three points on a coordinate plane. The x-axis ranges from negative 9 to 9 in unit increments and the y-axis ranges from negative 11 to 11 in unit increments.

Which table shows that y=−f(x)
results in a reflection over the x-axis?

(1 point)
Responses

x −f(x)
−2
0
−3
1
−6
2
x f left parenthesis x right parenthesis negative 2 0 negative 3 1 negative 6 2

x −f(x)
2 0
3 1
6 2
x f left parenthesis x right parenthesis 2 0 3 1 6 2

x −f(x)
2 0
3 −1
6 −2
x f left parenthesis x right parenthesis 2 0 3 negative 1 6 negative 2

x −f(x)
−2
0
−3
−1
−6
−2
x f left parenthesis x right parenthesis negative 2 0 negative 3 negative 1 negative 6 negative 2
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Answer 1

To determine which table correctly represents \( y = -f(x) \), we must understand that reflecting a function over the x-axis effectively means that the y-values of the function become their opposites. In other words, if a point on the original function is \((x, f(x))\), then under the transformation \(y = -f(x)\), that point would move to \((x, -f(x))\).

Let's examine each response:

1. **Response 1**:
- \( x \) | \( -f(x) \)
- -2 | ?
- 0 | ?
- -3 | ?
- 1 | ?
- -6 | ?
- 2 | ?

This response does not provide specific values for \( -f(x) \), making it impossible to verify the reflection.

2. **Response 2**:
- \( x \) | \( -f(x) \)
- 2 | 0
- 3 | 1
- 6 | 2

If this is the table for \( -f(x) \), the corresponding \( f(x) \) values need to be checked. The correct opposite values would be:
- For \( 2 \): \( f(2) = 0 \)
- For \( 3 \): \( f(3) = -1 \)
- For \( 6 \): \( f(6) = -2 \)
Hence, this arrangement does not represent a reflection.

3. **Response 3**:
- \( x \) | \( -f(x) \)
- 2 | 0
- 3 | -1
- 6 | -2

If \( -f(x) \) has the values given, then:
- For \( 2 \): \( f(2)\) = 0
- For \( 3 \): \( f(3) = 1\)
- For \( 6 \): \( f(6) = 2\)
This aligns correctly with the original values under the reflection rule.

4. **Response 4**:
- \( x \) | \( -f(x) \)
- -2 | ?
- 0 | ?
- -3 | ?
- -1 | ?
- -6 | ?
- -2 | ?

Again, not enough information is provided to determine if this is correct.

After analyzing all the responses, **Response 3** demonstrates the correct values for \( -f(x) \) after reflecting over the x-axis, since the output values for \( -f(x) \) are exactly the negatives of the input values of \( f(x) \).

Thus, the answer is:
**Response 3.**